CompleteBernsteinFunction-class | R Documentation |
A virtual superclass for all Bernstein functions which can represented
by a Stieltjes density (no drift or killing rate). That means that there
exists a Stieltjes measure \sigma
such that
\psi(x) = \int_0^\infty \frac{x}{x + u} \sigma(du) , x > 0 .
For continuous Stieltjes densities, the values of the Bernstein function
are calculated with stats::integrate()
by using the representation
\psi(x)
= \int_{0}^{\infty} x \mathrm{Beta}(1, x + u) \sigma(du), \quad x > 0 ,
and the values of the iterated differences are calculated by using the representation
(-1)^{j-1} \Delta^{j} \psi(x)
= \int_{0}^{\infty} u \mathrm{Beta}(j+1, x + u) \sigma(du) ,
\quad x > 0 .
For discrete Lévy densities
\sigma(du) = \sum_{i} y_i \delta_{u_i}(du)
,
the values of the Bernstein function are calculated by using the
representation
\psi(x)
= \sum_{i} x \mathrm{Beta}(1, x + u_i) y_i, \quad x > 0 ,
and the values of the iterated differences are calculated by using the representation
(-1)^{j-1} \Delta^{j} \psi(x)
= \sum_{i} u_i \mathrm{Beta}(j+1, x + u_i) y_i ,
\quad x > 0 .
levyDensity()
, stieltjesDensity()
, valueOf()
,
intensities()
, uexIntensities()
, exIntensities()
, exQMatrix()
,
rextmo()
, rpextmo()
Other Bernstein function classes:
AlphaStableBernsteinFunction-class
,
BernsteinFunction-class
,
CompositeScaledBernsteinFunction-class
,
ConstantBernsteinFunction-class
,
ConvexCombinationOfBernsteinFunctions-class
,
ExponentialBernsteinFunction-class
,
GammaBernsteinFunction-class
,
InverseGaussianBernsteinFunction-class
,
LevyBernsteinFunction-class
,
LinearBernsteinFunction-class
,
ParetoBernsteinFunction-class
,
PoissonBernsteinFunction-class
,
ScaledBernsteinFunction-class
,
SumOfBernsteinFunctions-class
Other Virtual Bernstein function classes:
BernsteinFunction-class
,
LevyBernsteinFunction-class
Other Levy Bernstein function classes:
AlphaStableBernsteinFunction-class
,
ExponentialBernsteinFunction-class
,
GammaBernsteinFunction-class
,
InverseGaussianBernsteinFunction-class
,
LevyBernsteinFunction-class
,
ParetoBernsteinFunction-class
,
PoissonBernsteinFunction-class
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